Numerical solutions to low and high-dimensional Allen–Cahn equations using stochastic differential equations and neural networks

Abstract

In this paper, we focus on solving semilinear parabolic differential equations in low and high-dimensional spaces by using backward stochastic differential equations and deep neural networks (the BSDE solver introduced by Han et al. in 2017). A 5D test problem was created to test the accuracy of the algorithm and to understand the key parameters in the neural network. In addition, we focus on Allen–Cahn equations in 1D, 3D, and 60D with different potential functions, including higher-order polynomial potential functions. To the best of our knowledge, this is the first time that Allen–Cahn equations were investigated in low and high dimensional spaces using the same algorithm. In addition, double well potential functions and higher order potential functions are also investigated using the same algorithm. Some patterns are observed through simulations with regard to the relations between the solutions and the order of potential functions and spatial dimensions.